Journal of Nanomaterials

Journal of Nanomaterials / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 2072789 | https://doi.org/10.1155/2019/2072789

Ye-Jun Ge, Jia-Bao Liu, Muhammad Younas, Muhammad Yousaf, Waqas Nazeer, "Analysis of and Nanotubes via Topological Indices", Journal of Nanomaterials, vol. 2019, Article ID 2072789, 10 pages, 2019. https://doi.org/10.1155/2019/2072789

Analysis of and Nanotubes via Topological Indices

Academic Editor: Stefano Bellucci
Received07 Nov 2018
Accepted24 Feb 2019
Published23 Apr 2019

Abstract

Scientists are creating materials, for example, a carbon nanotube-based composite created by NASA that bends when a voltage is connected. Applications incorporate the use of an electrical voltage to change the shape (transform) of air ship wings and different structures. Topological indices are numbers related with molecular graphs to allow quantitative structure activity/property/poisonous relationships. Topological indices catch symmetry of molecular structures and give it a scientific dialect to foresee properties, for example, boiling points, viscosity, and the radius of gyrations. We compute -polynomials of two nanotubes, and . The closed form of -polynomials for these nanotubes produces formulas of numerous degree-based topological indices which are functions relying on parameters of the structure and, in combination, decide properties of the concerned nanotubes. Moreover, we sketch our results by using Maple 2015 to see the dependence of our results on the involved parameters.

1. Introduction

The expression “nano” alludes to the metric prefix 10-9. It implies one billionth of something. “Nano” can be attributed to any unit of measure. For instance, you may report a little mass in nanograms or the measure of fluid in one cell as far as nanoliters. Nanoscience is the investigation of structures and materials on the size of nanometers. When structures are made enough in the nanometer measure range, they can go up against fascinating and valuable properties. Nanoscale structures have existed in nature well before researchers started considering them in labs. Researchers have even made nanostructures in the lab that copy a portion of nature’s stunning nanostructures [13].

Nanoscience has effectively affected our lives with developments, for example, stain-resistant fabrics propelled by nanoscale highlights found on lotus plants and PC hard drives, which store data on magnetic strips that are only 20 nanometers thick. Researchers from several disciplines material science, biology, chemistry, and physics utilize nanoscience standards for applications in computing, medication, energy, information storage, etc. In spite of the fact that achievements in any research field are hard to anticipate, the eventual fate of nanoscience will probably include scaling up from atomic get together and singular nanodevices to macroscopic systems and structures with developing properties and different capacities. Carbon nanotubes can be utilized as the pores in layers to run reverse osmosis desalination plants. Water atoms go through the smoother walls of carbon nanotubes more effortlessly than through different sorts of nanopores, which requires less power. Different scientists are utilizing carbon nanotubes to develop little, inexpensive water purification devices required in poor nations. Sensors utilizing carbon nanotube detection elements are fit for distinguishing a scope of chemical vapors. These sensors work by responding to the changes in the resistance of a carbon nanotube within the sight of a chemical vapor [47].

The features of energetics and electronic properties of carbon nanotubes, containing a pentagon-heptagon pair (5/7) topological defect in the hexagonal network of the zigzag configuration, are investigated using the extended Su-Schriffer-Heeger model based on the tight binding approximation in real space. Calculations show that this pentagon-heptagon pair defect in the nanotube structures is not only responsible for a change in nanotube diameter but also governs the electronic behavior around the Fermi level [8].

In nanoscience, (where and express the number of heptagons in each row and the number of periods in whole lattice, respectively) nanotube is a class of C5C7 net which is yielded by alternating C5 and C7. The standard tiling of C5 and C7 can cover either a cylinder or a torus, and each period of consisted of three rows (more details on th period can be referred to in Figure 1). H-Naphtalenic nanotubes (where and are denoted as the number of pairs of hexagons in first row and the number of alternative hexagons in a column, respectively) are a trivalent decoration with sequence of C6, C6, C4, C6, C6, C4, … in the first row and a sequence of C6, C8, C6, C8, … in the other rows. In other words, this nanolattice can be considered as a plane tiling of C4, C6, and C8. Therefore, this class of tiling can cover either a cylinder or a torus [9].

In the field of chemical graph theory, the molecular graph is a simple connected graph in which atoms are taken as vertices and chemical bonds are taken as edges. The vertex set is usually denoted by , and the edge set is denoted by . The degree of a vertex is denoted by , which is the total number of vertices attached with it. The valence in chemistry and the degree of a vertex in a graph are closely related. For details about graph theory and its notion, we refer to the book [10].

Another emerging field is cheminformatics. In cheminformatics, the quantitative structure-activity (QSAR) and the structure-property (QSPR) relationship foresees the biological activity and properties of understudy nanomaterial. In these investigations, topological indices and some physicochemical properties are utilized to foresee bioactivity of understudy nanomaterial, see [1115]. Polynomials have additionally valuable applications in chemical graph theory, for example, Hosoya polynomial (also called Wiener polynomial) [16] which assumes an indispensable job in deciding distance-based topological indices. Among other polynomials, -polynomial [17] presented in 2015 assumes a similar job in deciding the closed form of many degree-based topological indices [1822]. The primary favorable position of -polynomial is the abundance of data that it contains about degree-based graph invariants.

In this paper, we focus on the degree-based combinatorial facts of two nanotubes. At first, we give -polynomial of these nanotubes shown in Figures 1 and 2. Then, by using calculus rules on the -polynomials, we recover nine topological indices.

2. Basic Definition and Literature Review

In this section, we give basic definitions and notions. Throughout this paper, we fix the following notion. (1) simple graph(2) of vertices of (3) of edges(4) of elements in (5) = numbers of elements in (6) = degree of vertex

Definition 1 (see [17]). For a simple connected graph , the -polynomial is defined as follows: where , , and is the edge such that .

In 1947, the first topological index was introduced by Wiener, when he was studying the boiling point of alkanes [23]. Subsequently, Wiener established the framework of the topological index [24, 25].

After Wiener, in 1975, Milan Randić introduced the Randić index in [26], which is defined as and is one of the oldest degree-based topological index. In the year 1998, researchers in [27, 28] introduced the generalized version of the Randić index. The generalized Randić index gets a great attention from a mathematician [29]. We refer to [30] for mathematical properties of this index, and for detailed survey, we refer to [31]. The general Randić index is defined as and the generalized inverse Randić index is defined as

If we take in the generalized Randić index, we get . The Randić index is likewise the most mainstream regularly connected and most concentrated among all other topological indices. Numerous papers and books, for example, [3234], are written on this topological index. Randić himself wrote two surveys on his Randić index [35, 36], and there are three more surveys [3739].

The first and second Zagreb indices are defined by Gutman et al. as

For insight about these indices, we refer [4044] to the readers. Both the first Zagreb index and the second Zagreb index give more weights to the internal vertices and edges and less weights to external vertices and edges which contradict instinctive thinking. Consequently, they were modified in [45] as the second modified Zagreb index:

The symmetric division index () is one of the 148 discrete Adriatic indices and is a decent indicator of the aggregate surface area for polychlorobiphenyls [46]. The symmetric division index of a graph is defined as

Harmonic index is defined as

As far as we know, this index firstly appeared in [47] and studied in [48].

The inverse sum index is the descriptor that was chosen in [49] as a huge indicator of the aggregate surface area of octane isomers and for which the extremal graphs acquired have an especially straightforward and rich structure. It is denoted by and is defined as

In [50], Furtula et al. defined the augmented Zagreb index as

This graph invariant has wound up being a critical judicious index in the examination of the heat formation in heptanes and octanes (see [50]), whose forecast control is better than the atomic bond connectivity (ABC) index [5153]. The relationship between -polynomial and indices is presented in Table 1 [17].


Topological indexDerivation from


and

3. Main Results

In this section, we give our computational results.

3.1. -Polynomial and Degree-Based Topological Indices of

Theorem 1. The -polynomial of nanotube is

Proof 1. Then, the vertex set of has the following two partitions with respect to degree: The edge set of has three partitions based on the degree of end vertices: such that Now,

Figure 3 shows the behavior of -polynomial of onto the involved parameters .

Proposition 2. For the nanotube , we have

Proof 2. Let Then, Now from Table 1, (1)First Zagreb index (2)Second Zagreb index (3)Modified second Zagreb index (4)Generalized Randić index (5)Inverse Randić index (6)Symmetric division index (7)Harmonic index (8)Inverse sum index (9)Augmented Zagreb index

3.2. The -Polynomial and Degree-Based Topological Indices of

Theorem 3. For the H-naphtalenic nanotubes , we have

Proof 3. The edge set of has the following two partitions with respect to degree of the end vertices: such that Now,

Figure 4 shows the behavior of -polynomial of , onto the involved parameters .

Proposition 4. For the H-naphtalenic nanotubes , we have

Proof 4. Let Then, (1)First Zagreb index (2)Second Zagreb index (3)Modified second Zagreb index (4)Generalized Randić index (5)Inverse Randić index (6)Symmetric division index (7)Harmonic index (8)Inverse sum index (9)Augmented Zagreb index

4. Discussion and Concluding Remarks

We computed -polynomials of and . From these -polynomials, we recovered nine degree-based topological indices by applying fundamental rules of calculus. Our calculated results help to understand topology of understudy nanotubes. For instant, the first Zagreb index is used to known about pi-electronic energy. Augmented Zagreb index can be used in the investigation of the heat of formation. Randić index is the most applied and investigated degree-based topological index. Figures 513 show the dependence of our results on the involved parameters. Blue color is fixed for , and green color is fixed for .

Data Availability

All data is included within this paper.

Conflicts of Interest

The authors do not have any competing interests.

Authors’ Contributions

All authors contribute equally in this paper.

Acknowledgments

The work was supported in part by funding. This research was funded by the Natural Science Research Key Project from Education Department of Anhui Province (Grant No. KJ2017A492) and youth research special fund project of Anhui Jianzhu University (Grant No. 2011183-8).

References

  1. H. Deng, X. Li, Q. Peng, X. Wang, J. Chen, and Y. Li, “Monodisperse magnetic single-crystal ferrite microspheres,” Angewandte Chemie, vol. 117, no. 18, pp. 2842–2845, 2005. View at: Publisher Site | Google Scholar
  2. G. A. Somorjai and Y. G. Borodko, “Research in nanosciences–great opportunity for catalysis science,” Catalysis Letters, vol. 76, no. 1/2, pp. 1–5, 2001. View at: Publisher Site | Google Scholar
  3. R. J. P. Corriu, A. Mehdi, and C. Reyé, “Molecular chemistry and nanosciences: on the way to interactive materials,” Journal of Materials Chemistry, vol. 15, no. 40, pp. 4285–4294, 2005. View at: Publisher Site | Google Scholar
  4. R. H. Baughman, A. A. Zakhidov, and W. A. De Heer, “Carbon nanotubes--the route toward applications,” Science, vol. 297, no. 5582, pp. 787–792, 2002. View at: Publisher Site | Google Scholar
  5. B. S. Harrison and A. Atala, “Carbon nanotube applications for tissue engineering,” Biomaterials, vol. 28, no. 2, pp. 344–353, 2007. View at: Publisher Site | Google Scholar
  6. M. F. L. De Volder, S. H. Tawfick, R. H. Baughman, and A. J. Hart, “Carbon nanotubes: present and future commercial applications,” Science, vol. 339, no. 6119, pp. 535–539, 2013. View at: Publisher Site | Google Scholar
  7. A. K. T. Lau and D. Hui, “The revolutionary creation of new advanced materials—carbon nanotube composites,” Composites Part B: Engineering, vol. 33, no. 4, pp. 263–277, 2002. View at: Publisher Site | Google Scholar
  8. H. Hui-Fang, L. Yi-bing, Z. Wei-ya, and T. Dong-sheng, “Electronic properties of carbon nanotubes with a pentagon-heptagon pair defect,” Chinese Physics, vol. 10, no. 6, pp. 531–536, 2001. View at: Publisher Site | Google Scholar
  9. W. Gao, W. Wang, M. K. Jamil, and M. R. Farahani, “Electron energy studying of molecular structures via forgotten topological index computation,” Journal of Chemistry, vol. 2016, 7 pages, 2016. View at: Publisher Site | Google Scholar
  10. D. B. West, An Introduction to Graph Theory, Prentice-Hall, 1996.
  11. G. Rucker and C. Rucker, “On topological indices, boiling points, and cycloalkanes,” Journal of Chemical Information and Computer Sciences, vol. 39, no. 5, pp. 788–802, 1999. View at: Publisher Site | Google Scholar
  12. S. Klavžar and I. Gutman, “A comparison of the Schultz molecular topological index with the Wiener index,” Journal of Chemical Information and Computer Sciences, vol. 36, no. 5, pp. 1001–1003, 1996. View at: Publisher Site | Google Scholar
  13. F. M. Brückler, T. Došlić, A. Graovac, and I. Gutman, “On a class of distance-based molecular structure descriptors,” Chemical Physics Letters, vol. 503, no. 4-6, pp. 336–338, 2011. View at: Publisher Site | Google Scholar
  14. H. Deng, J. Yang, and F. Xia, “A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3017–3023, 2011. View at: Publisher Site | Google Scholar
  15. H. Zhang and F. Zhang, “The Clar covering polynomial of hexagonal systems I,” Discrete Applied Mathematics, vol. 69, no. 1-2, pp. 147–167, 1996. View at: Publisher Site | Google Scholar
  16. I. Gutman, “Some properties of the Wiener polynomials,” Graph Theory Notes New York, vol. 125, pp. 13–18, 1993. View at: Google Scholar
  17. E. Deutsch and S. Klavzar, “M-polynomial, and degree-based topological indices,” Iranian Journal of Mathematical Chemistry, vol. 6, pp. 93–102, 2015. View at: Google Scholar
  18. M. Munir, W. Nazeer, S. Rafique, and S. M. Kang, “M-polynomial and related topological indices of nanostar dendrimers,” Symmetry, vol. 8, no. 9, p. 97, 2016. View at: Publisher Site | Google Scholar
  19. M. Munir, W. Nazeer, S. Rafique, A. R. Nizami, and S. M. Kang, “M-polynomial and degree-based topological indices of titania nanotubes,” Symmetry, vol. 8, no. 11, p. 117, 2016. View at: Publisher Site | Google Scholar
  20. M. Munir, W. Nazeer, S. Rafique, and S. M. Kang, “M-polynomial and degree-based topological indices of polyhex nanotubes,” Symmetry, vol. 8, no. 12, p. 149, 2016. View at: Publisher Site | Google Scholar
  21. M. R. Farahani, M. K. Jamil, and M. Imran, “Vertex PIv topological index of titania carbon nanotubes TiO2 (m, n),” Applied Mathematics and Nonlinear Sciences, vol. 1, no. 1, pp. 175–2845, 2016. View at: Publisher Site | Google Scholar
  22. M. Munir, W. Nazeer, S. Shahzadi, and S. M. Kang, “Some invariants of circulant graphs,” Symmetry, vol. 8, no. 11, p. 134, 2016. View at: Publisher Site | Google Scholar
  23. H. Wiener, “Structural determination of paraffin boiling points,” Journal of the American Chemical Society, vol. 69, no. 1, pp. 17–20, 1947. View at: Publisher Site | Google Scholar
  24. A. A. Dobrynin, R. Entringer, and I. Gutman, “Wiener index of trees: theory and applications,” Acta Applicandae Mathematica, vol. 66, no. 3, pp. 211–249, 2001. View at: Publisher Site | Google Scholar
  25. I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag New York, New York, NY, USA, 1986.
  26. M. Randić, “Characterization of molecular branching,” Journal of the American Chemical Society, vol. 97, no. 23, pp. 6609–6615, 1975. View at: Publisher Site | Google Scholar
  27. B. Bollobás and P. Erdös, “Graphs of extremal weights,” Ars Combinatoria, vol. 50, pp. 225–233, 1998. View at: Google Scholar
  28. D. Amić, D. Beslo, B. Lučić, S. Nikolić, and N. Trinajstić, “The vertex-connectivity index revisited,” Journal of Chemical Information and Computer Sciences, vol. 38, no. 5, pp. 819–822, 1998. View at: Publisher Site | Google Scholar
  29. Y. Hu, X. Li, Y. Shi, T. Xu, and I. Gutman, “On molecular graphs with smallest and greatest zeroth-order general Randić index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 54, pp. 425–434, 2005. View at: Google Scholar
  30. G. Caporossi, I. Gutman, P. Hansen, and L. Pavlovic, “Graphs with maximum connectivity index,” Computational Biology and Chemistry, vol. 27, no. 1, pp. 85–90, 2003. View at: Publisher Site | Google Scholar
  31. M. Alaeiyan, M. R. Farahani, and M. K. Jamil, “Computation of the fifth geometric-arithmetic index for polycyclic aromatic hydrocarbons PAHk,” Applied Mathematics and Nonlinear Sciences, vol. 1, no. 1, pp. 283–290, 2016. View at: Publisher Site | Google Scholar
  32. L. B. Kier and L. H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New York, 1976.
  33. L. B. Kier and L. H. Hall, Molecular Connectivity in Structure-Activity Analysis, Wiley, New York, 1986.
  34. X. Li and I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors, University of Kragujevac, Kragujevac, 2006.
  35. M. Randić, “On history of the Randić index and emerging hostility toward chemical graph theory,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 59, pp. 5–124, 2008. View at: Google Scholar
  36. M. Randić, “The connectivity index 25 years after,” Journal of Molecular Graphics & Modelling, vol. 20, no. 1, pp. 19–35, 2001. View at: Publisher Site | Google Scholar
  37. I. Gutman and B. Furtula, Recent Results in the Theory of Randić Index, University of Kragujevac, Kragujevac, 2008.
  38. X. Li and Y. Shi, “A survey on the Randić index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 59, pp. 127–156, 2008. View at: Google Scholar
  39. X. Li, Y. Shi, and L. Wang, Recent Results in the Theory of Randić Index, I. Gutman and B. Furtula, Eds., University of Kragujevac, Kragujevac, 2008.
  40. S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, “The Zagreb indices 30 years after,” Croatica Chemica Acta, vol. 76, pp. 113–124, 2003. View at: Google Scholar
  41. I. Gutman and K. C. Das, “The first Zagreb indices 30 years after,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 50, pp. 83–92, 2004. View at: Google Scholar
  42. K. Das and I. Gutman, “Some properties of the second Zagreb index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 52, pp. 103–112, 2004. View at: Google Scholar
  43. N. Trinajstić, S. Nikolić, A. Miličević, and I. Gutman, “On Zagreb indices,” Kemija u Industriji, vol. 59, pp. 577–589, 2010. View at: Google Scholar
  44. D. Kovačević and A. Graovac, “Valence connectivities versus Randić, Zagreb and modified Zagreb index: a linear algorithm to check discriminative properties of indices in acyclic molecular graphs,” Croatica Chemica Acta, vol. 77, pp. 501–508, 2004. View at: Google Scholar
  45. A. Miličević, S. Nikolić, and N. Trinajstić, “On reformulated Zagreb indices,” Molecular Diversity, vol. 8, no. 4, pp. 393–399, 2004. View at: Publisher Site | Google Scholar
  46. C. K. Gupta, V. Lokesha, S. B. Shwetha, and P. S. Ranjini, “On the symmetric division deg index of graph,” Southeast Asian Bulletin of Mathematics, vol. 40, no. 1, pp. 59–80, 2016, 22p. View at: Google Scholar
  47. W. Gao and M. R. Farahani, “Degree-based indices computation for special chemical molecular structures using edge dividing method,” Applied Mathematics and Nonlinear Sciences, vol. 1, no. 1, pp. 99–122, 2016. View at: Publisher Site | Google Scholar
  48. O. Favaron, M. Mahéo, and J. F. Saclé, “Some eigenvalue properties in graphs (conjectures of Graffiti—II),” Discrete Mathematics, vol. 111, no. 1-3, pp. 197–220, 1993. View at: Publisher Site | Google Scholar
  49. A. T. Balaban, “Highly discriminating distance based numerical descriptor,” Chemical Physics Letters, vol. 89, pp. 399–404, 1982. View at: Google Scholar
  50. B. Furtula, A. Graovac, and D. Vukičević, “Augmented Zagreb index,” Journal of Mathematical Chemistry, vol. 48, no. 2, pp. 370–380, 2010. View at: Publisher Site | Google Scholar
  51. K. C. Das, “Atom–bond connectivity index of graphs,” Discrete Applied Mathematics, vol. 158, no. 11, pp. 1181–1188, 2010. View at: Publisher Site | Google Scholar
  52. E. Estrada, L. Torres, L. Rodriguez, and I. Gutman, “An atom–bond connectivity index: modeling the enthalpy of formation of alkanes,” Indian Journal of Chemistry, vol. 37A, pp. 849–855, 1998. View at: Google Scholar
  53. E. Estrada, “Atom–bond connectivity and the energetic of branched alkanes,” Chemical Physics Letters, vol. 463, no. 4-6, pp. 422–425, 2008. View at: Publisher Site | Google Scholar

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